Welcome dear students! Today we are going to learn about Comparing Quantities from Class 7 Maths.
Let us begin with an interesting situation. Anita scored 320 marks out of 400, while Rita scored 300 marks out of 360. Anita claimed she did better because her total marks were higher. Do you agree with her? Who do you think has done better? Mansi pointed out that we cannot decide just by comparing total marks because the maximum marks are different. She suggested looking at the percentages. Anita's percentage was 80, and Rita's percentage was 83.3. This clearly shows Rita has done better. Do you agree? Percentages are numerators of fractions with denominator 100 and have been used in comparing results. Let us try to understand this in detail.
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Section 7.1.1 covers the Meaning of Percentage. Percent is derived from the Latin word per centum, meaning per hundred. Percent is represented by the symbol % and means hundredths too. That is 1% means 1 out of hundred or one hundredth. It can be written as: 1% = 1/100 = 0.01. To understand this, let us consider an example. Rina made a table top of 100 different coloured tiles. She counted yellow, green, red and blue tiles separately. Yellow has 14 tiles, which is 14 per hundred, written as 14/100, read as 14%. Green has 26 tiles, written as 26/100, read as 26%. Red has 35 tiles, written as 35/100, read as 35%. Blue has 25 tiles, written as 25/100, read as 25%. The total is 100 tiles.
Now let us look at the first Try These exercise. Find the percentage of children of different heights. The data lists heights 110 cm, 120 cm, 128 cm, and 130 cm with 22, 25, 32, and 21 children respectively. Since the total is 100, the fraction for each is simply the number of children over 100, and the percentage matches the numerator. So, 110 cm is 22%, 120 cm is 25%, 128 cm is 32%, and 130 cm is 21%. The second Try These asks about shoe pairs. Size 2 has 20 pairs, Size 3 has 30 pairs, Size 4 has 28 pairs, Size 5 has 14 pairs, and Size 6 has 8 pairs. The total is 100 pairs. Writing this in tabular form, the percentages are 20%, 30%, 28%, 14%, and 8% respectively.
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In all these examples, the total number of items adds up to 100. But how do we calculate percentage if the total does not add up to 100? We need to convert the fraction to an equivalent fraction with denominator 100. Consider a necklace with 20 beads in two colours. Red beads are 8, fraction is 8/20. To get denominator 100, we multiply numerator and denominator by 5: 8/20 × 5/5 = 40/100 = 40%. Blue beads are 12, fraction is 12/20. Multiplying by 5/5 gives 12/20 × 5/5 = 60/100 = 60%. Total is 20 beads. We see three methods can be used. The table method multiplies the fraction by 100/100. Anwar used the unitary method: Out of 20 beads, 8 are red. Hence, out of 100, the number of red beads is 8/20 × 100 = 40 = 40%. Asha multiplied 8/20 by 5/5 to get 40/100 = 40%. You can use whichever method suits you. The method used by Anwar works for all ratios. Can Asha's method work for all ratios? Asha's method works only if you can find a natural number which, when multiplied with the denominator, gives 100. Since the denominator was 20, she multiplied by 5. If the denominator was 6, she could not use this method. Do you agree?
Let us look at the next Try These. The textbook shows a collection of 10 chips with different colours: 4 Green, 3 Red, and 3 Blue. Fill the table to find the percentage of each colour. Green is 4/10, which is 40/100 or 40%. Red is 3/10, which is 30/100 or 30%. Blue is 3/10, which is 30/100 or 30%. The second Try These: Mala has 20 gold bangles and 10 silver bangles. Total is 30 bangles. Percentage of gold bangles = 20/30 × 100 = 2000/30 = 66⅔%. Percentage of silver bangles = 10/30 × 100 = 1000/30 = 33⅓%.
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THINK, DISCUSS AND WRITE. First, look at the examples. In the atmosphere, 1 g of air contains 0.78 g Nitrogen, 0.21 g Oxygen, and 0.01 g Other gas. Alternatively, it is written as 78% Nitrogen, 21% Oxygen, and 1% Other gas. Which form is better for comparison? Percentages are much clearer and easier to compare. Second, a shirt has 3/5 Cotton and 2/5 Polyester, or 60% Cotton and 40% Polyester. Discuss with your classmates which representation is more useful. Percentages again provide an immediate understanding of proportions.
Section 7.1.2 covers Converting Fractional Numbers to Percentage. Fractional numbers can have different denominators. To compare them, we convert to percentages. Example 1: Write 1/3 as percent. Solution: We have 1/3 = 1/3 × 100/100 = 100/3 % = 33⅓%. Example 2: Out of 25 children in a class, 15 are girls. What is the percentage of girls? Solution: Out of 25 children, there are 15 girls. Therefore, percentage of girls = 15/25 × 100 = 60. There are 60% girls in the class. Example 3: Convert 5/4 to percent. Solution: We have 5/4 = 5/4 × 100/100 = 500/4 % = 125%. From these examples, we find that the percentages related to proper fractions are less than 100, whereas percentages related to improper fractions are more than 100.
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THINK, DISCUSS AND WRITE: Can you eat 50% of a cake? Yes. Can you eat 100% of a cake? Yes. Can you eat 150% of a cake? No, because you cannot eat more than the whole cake. Can a price of an item go up by 50%? Yes. Can a price of an item go up by 100%? Yes, it doubles. Can a price of an item go up by 150%? Yes, it becomes two and a half times the original. Discuss these questions to understand the real-world meaning of percentages.
Section 7.1.3 covers Converting Decimals to Percentage. Example 4: Convert the given decimals to percents. (a) 0.75 = 75/100 × 100% = 75%. (b) 0.09 = 9/100 × 100% = 9%. (c) 0.2 = 2/10 × 100% = 20%. Try These 1: Convert the following to percents. (a) 12/16 = 3/4 = 3/4 × 100% = 75%. (b) 3.5 = 35/10 = 350/100 = 350%. (c) 49/50 = 49/50 × 100% = 98%. (d) 2/2 = 1 = 100%. (e) 0.05 = 5/100 × 100% = 5%. Try These 2: (i) Out of 32 students, 8 are absent. Percent absent = 8/32 × 100 = 1/4 × 100 = 25%. (ii) There are 25 radios, 16 out of order. Percent out of order = 16/25 × 100 = 64%. (iii) A shop has 500 items, 5 defective. Percent defective = 5/500 × 100 = 1%. (iv) There are 120 voters, 90 voted yes. Percent voted yes = 90/120 × 100 = 3/4 × 100 = 75%.
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Section 7.1.4 covers Converting Percentages to Fractions or Decimals. We can do the reverse. The textbook provides a conversion table. Let us observe it carefully. 1% is 1/100 as a fraction and 0.01 as a decimal. 10% is 10/100, which simplifies to 1/10, and 0.10 as a decimal. 25% is 25/100, which simplifies to 1/4, and 0.25 as a decimal. 50% is 50/100, which simplifies to 1/2, and 0.50 as a decimal. 90% is 90/100, which simplifies to 9/10, and 0.90 as a decimal. 125% is 125/100, which simplifies to 5/4, and 1.25 as a decimal. 250% is 250/100, which simplifies to 5/2, and 2.50 as a decimal.
Parts always add to give a whole. When we add percentages, we get 100. If 30% are boys, then girls are 100 – 30 = 70%. Try These 1: Fill in the blanks. 35% + 65% = 100%. 64% + 20% + 16% = 100%. 45% = 100% – 55%. 70% = 100% – 30%. Try These 2: If 65% of students in a class have a bicycle, what percent do not have bicycles? 100% – 65% = 35%. Try These 3: A basket has apples, oranges and mangoes. 50% are apples, 30% are oranges. What percent are mangoes? 100% – (50% + 30%) = 20%. THINK, DISCUSS AND WRITE: Consider expenditure on a dress: 20% on embroidery, 50% on cloth, 30% on stitching. These add to 100%. You can think of similar examples like monthly budget or food ingredients.
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Section 7.1.5 Fun with Estimation. Percentages help us to estimate the parts of an area. Example 5: What percent of the adjoining figure is shaded? Solution: We first find the fraction of the figure that is shaded. From this fraction, the percentage of the shaded part can be found. You will find that half of the figure is shaded. And, 1/2 = 1/2 × 100% = 50%. Thus, 50% of the figure is shaded. Now, look at the Try These. The textbook shows two tangram figures and asks you to estimate what percent of each figure is shaded. You can also make some more figures yourself and ask your friends to estimate the shaded parts.
Section 7.2 covers Use of Percentages. Section 7.2.1 Interpreting Percentages. We saw how percentages were helpful in comparison. We have also learnt to convert fractional numbers and decimals to percentages. Now, we shall learn how percentages can be used in real life. For this, start with interpreting the following statements: 5% of the income is saved by Ravi. 20% of Meera’s dresses are blue in colour. Rekha gets 10% on every book sold by her. What can you infer from each of these statements? By 5% we mean 5 parts out of 100 or we write it as 5/100. It means Ravi is saving 5 rupees out of every 100 rupees that he earns. In the same way, 20% of Meera's dresses being blue means 20 out of every 100 dresses are blue. Rekha getting 10% means she earns 10 rupees commission for every 100 rupees of book sales.
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Section 7.2.2 Converting Percentages to “How Many”. Consider the following examples: Example 6: A survey of 40 children showed that 25% liked playing football. How many children liked playing football? Solution: Total is 40. Arun's method: Out of 100, 25 like football. So out of 40, number is 25/100 × 40 = 10. Meena's method: 25% of 40 = 25/100 × 40 = 10. Hence, 10 children out of 40 like playing football. Try These 1: Find: (a) 50% of 164 = 1/2 × 164 = 82. (b) 75% of 12 = 3/4 × 12 = 9. (c) 12½% of 64 = 25/200 × 64 = 8. Try These 2: 8% children of a class of 25 like getting wet in the rain. How many children? 8/100 × 25 = 2 children.
Example 7: Rahul bought a sweater and saved 200 rupees when a discount of 25% was given. What was the price of the sweater before the discount? Solution: Rahul has saved 200 rupees when price of sweater is reduced by 25%. This means that 25% reduction in price is the amount saved by Rahul. Mohan's solution: 25% of the original price = 200. 25/100 × P = 200. P/4 = 200. P = 800 rupees. Abdul's solution: 25 rupees is saved for every 100 rupees. Amount for which 200 rupees is saved = 100/25 × 200 = 800 rupees. Both obtained the original price of sweater as 800 rupees. Try These: 1. 9 is 25% of what number? 25/100 × X = 9. X/4 = 9. X = 36. 2. 75% of what number is 15? 75/100 × X = 15. 3X/4 = 15. X = 20.
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Exercise 7.1 Solutions: Question 1: Convert the given fractional numbers to percents. (a) 1/8 = 1/8 × 100% = 12.5%. (b) 5/4 = 5/4 × 100% = 125%. (c) 3/40 = 3/40 × 100% = 7.5%. (d) 2/7 = 2/7 × 100% = 200/7 % = 28 4/7 %. Question 2: Convert the given decimal fractions to percents. (a) 0.65 = 65/100 × 100% = 65%. (b) 2.1 = 21/10 × 100% = 210%. (c) 0.02 = 2/100 × 100% = 2%. (d) 12.35 = 1235/100 × 100% = 1235%. Question 3: Estimate what part of the figures is coloured and hence find the percent which is coloured. (i) 1/4 is coloured = 25%. (ii) 3/8 is coloured = 37.5%. (iii) 1/8 is coloured = 12.5%. Question 4: Find: (a) 15% of 250 = 15/100 × 250 = 37.5. (b) 1% of 1 hour = 1/100 × 60 minutes = 0.6 minutes = 36 seconds. (c) 20% of 2500 rupees = 20/100 × 2500 = 500 rupees. (d) 75% of 1 kg = 75/100 × 1000 g = 750 g. Question 5: Find the whole quantity if: (a) 5% of X = 600. X = 600 × 100/5 = 12000. (b) 12% of X = 1080 rupees. X = 1080 × 100/12 = 9000 rupees. (c) 40% of X = 500 km. X = 500 × 100/40 = 1250 km. (d) 70% of X = 14 minutes. X = 14 × 100/70 = 20 minutes. (e) 8% of X = 40 litres. X = 40 × 100/8 = 500 litres.
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Question 6: Convert given percents to decimal fractions and also to fractions in simplest forms: (a) 25% = 0.25 = 1/4. (b) 150% = 1.5 = 3/2. (c) 20% = 0.20 = 1/5. (d) 5% = 0.05 = 1/20. Question 7: In a city, 30% are females, 40% are males. Percent children = 100% – (30% + 40%) = 30%. Question 8: Out of 15,000 voters, 60% voted. Percent who did not vote = 100% – 60% = 40%. Number who did not vote = 40/100 × 15000 = 6000. Question 9: Meeta saves 4000 rupees, which is 10% of her salary. 10/100 × Salary = 4000. Salary = 4000 × 10 = 40000 rupees. Question 10: A cricket team played 20 matches, won 25%. Matches won = 25/100 × 20 = 5 matches.
Section 7.2.3 Ratios to Percents. Example 8: Idli mixture is 2 parts rice, 1 part urad dal. Ratio is 2:1. Total parts 3. Rice is 2/3 part. Percentage of rice = 2/3 × 100 = 200/3 % = 66⅔%. Urad dal is 1/3 part. Percentage = 1/3 × 100 = 100/3 % = 33⅓%. Example 9: Divide 250 rupees in ratio 2:3:5. Total parts 10. Ravi gets 2/10 × 250 = 50 rupees. Percentage = 2/10 × 100 = 20%. Raju gets 3/10 × 250 = 75 rupees. Percentage = 3/10 × 100 = 30%. Roy gets 5/10 × 250 = 125 rupees. Percentage = 5/10 × 100 = 50%. Try These: 1. Divide 15 sweets between Manu and Sonu so they get 20% and 80%. Manu gets 20/100 × 15 = 3 sweets. Sonu gets 80/100 × 15 = 12 sweets. 2. Angles of a triangle are in ratio 2:3:4. Find each angle. Total parts = 2+3+4 = 9. Sum of angles = 180°. One part = 180/9 = 20°. Angles are 2×20° = 40°, 3×20° = 60°, 4×20° = 80°.
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Section 7.2.4 Increase or Decrease as Percent. Example 10: Won 6 games this year, 4 last year. Increase = 6 - 4 = 2. Percentage increase = amount of change / original amount × 100 = 2/4 × 100 = 50%. Example 11: Illiterate persons decreased from 150 lakhs to 100 lakhs. Decrease = 150 - 100 = 50 lakhs. Percentage decrease = 50/150 × 100 = 33⅓ %. Try These: 1. Price of shirt decreased from 280 to 210 rupees. Decrease = 70 rupees. Percent decrease = 70/280 × 100 = 25%. Marks increased from 20 to 30. Increase = 10. Percent increase = 10/20 × 100 = 50%. 2. Petrol was 1 rupee a litre, now 52. Increase = 51. Percent increase = 51/1 × 100 = 5100%.
Section 7.3 Prices Related to an Item. Buying price is Cost Price (CP). Selling price is Selling Price (SP). If CP < SP, profit = SP – CP. If CP = SP, no profit no loss. If CP > SP, loss = CP – SP. Example: Toy CP 72 rupees, SP 80 rupees. Profit 8 rupees. T-shirt CP 120 rupees, SP 100 rupees. Loss 20 rupees. Cycle CP 800 rupees, SP 940 rupees. Profit 140 rupees. 7.3.1 Profit or Loss as a Percentage. Always calculated on CP. For the toy: Profit % = Profit/CP × 100 = 8/72 × 100 = 100/9 = 11⅑ %. For the T-shirt: Loss % = Loss/CP × 100 = 20/120 × 100 = 100/6 = 16⅔ %. Try These 1: Chair CP 375 rupees, SP 400 rupees. Profit = 25 rupees. Gain % = 25/375 × 100 = 6⅔ %. Try These 2: CP 50 rupees, profit 12%. Profit = 12/100 × 50 = 6 rupees. SP = 50 + 6 = 56 rupees. Try These 3: An article was sold for 250 rupees with a profit of 5%. What was its cost price? Try These 4: An item was sold for 540 rupees at a loss of 5%. What was its cost price? I encourage you to pause and solve these last two questions yourself to practice finding the cost price when selling price and profit or loss percent are given.
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Example 12: Vase CP 120 rupees, loss 10%. Find SP. Sohan's method: Loss is 10% of 120 = 10/100 × 120 = 12 rupees. SP = CP – Loss = 120 – 12 = 108 rupees. Anandi's method: If CP is 100, SP is 90. For CP 120, SP = 90/100 × 120 = 108 rupees. Example 13: Toy SP 540 rupees, profit 20%. Find CP. Amina's method: 20% profit means if CP is 100, SP is 120. When SP is 540, CP = 100/120 × 540 = 450 rupees. Arun's method: 540 = CP + 20/100 × CP = 6/5 × CP. CP = 540 × 5/6 = 450 rupees.
Section 7.4 Charge Given on Borrowed Money or Simple Interest. Money borrowed is Principal. Extra money paid is Interest. Amount = Principal + Interest. Interest is given per cent per annum. 10% p.a. means on every 100 rupees borrowed, 10 rupees interest for one year. Example 14: Loan 5000 rupees at 15% per year. Interest for one year = 15/100 × 5000 = 750 rupees. Amount = 5000 + 750 = 5750 rupees. General formula: Interest = P × R / 100. 7.4.1 Interest for Multiple Years. Simple interest keeps principal unchanged. Interest for T years = P × R × T / 100. Amount = P + I. Try These: 1. 10000 rupees at 5% p.a. Interest = 10000 × 5 × 1 / 100 = 500 rupees. 2. 3500 rupees at 7% p.a. for two years. Interest = 3500 × 7 × 2 / 100 = 490 rupees. 3. 6050 rupees at 6.5% p.a. for 3 years. Interest = 6050 × 6.5 × 3 / 100 = 1179.75 rupees. Amount = 6050 + 1179.75 = 7229.75 rupees. 4. 7000 rupees at 3.5% p.a. for 2 years. Interest = 7000 × 3.5 × 2 / 100 = 490 rupees. Amount = 7000 + 490 = 7490 rupees.
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Example 15: Interest 750 rupees for 2 years on 4500 rupees. Find rate. Solution 1: 750 = 4500 × R × 2 / 100. 750 = 90 × R. R = 750/90 = 8⅓ %. Solution 2: For 2 years, interest 750 rupees. For 1 year, 375 rupees. On 4500 rupees, interest 375 rupees. On 100 rupees, rate = 375/4500 × 100 = 8⅓ %. Try These: 1. 2400 rupees at 5%. After how many years earn 240 interest? 240 = 2400 × 5 × T / 100. 240 = 120 × T. T = 2 years. 2. Interest 450 after 3 years at 5%. Find the sum. 450 = P × 5 × 3 / 100. 450 = 15P/100. P = 450 × 100/15 = 3000 rupees.
Exercise 7.2 Solutions: Question 1: Tell what is the profit or loss. Also find profit per cent or loss percent. (a) Shears CP 250, SP 325. Profit = 75. Profit % = 75/250 × 100 = 30%. (b) Refrigerator CP 12000, SP 13500. Profit = 1500. Profit % = 1500/12000 × 100 = 12.5%. (c) Cupboard CP 2500, SP 3000. Profit = 500. Profit % = 500/2500 × 100 = 20%. (d) Skirt CP 250, SP 150. Loss = 100. Loss % = 100/250 × 100 = 40%. Question 2: Convert each part of the ratio to percentage: (a) 3:1 -> 3/4 = 75%, 1/4 = 25%. (b) 2:3:5 -> 2/10 = 20%, 3/10 = 30%, 5/10 = 50%. (c) 1:4 -> 1/5 = 20%, 4/5 = 80%. (d) 1:2:5 -> 1/8 = 12.5%, 2/8 = 25%, 5/8 = 62.5%. Question 3: Population decreased from 25000 to 24500. Decrease = 500. Percent decrease = 500/25000 × 100 = 2%. Question 4: Car 350000 to 370000. Increase = 20000. Percent increase = 20000/350000 × 100 = 40/7 % = 5 5/7 %. Question 5: TV CP 10000, profit 20%. SP = 10000 + 20/100 × 10000 = 12000 rupees. Question 6: Washing machine SP 13500, loss 20%. SP = 80% of CP. CP = 13500 × 100/80 = 16875 rupees.
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Question 7: (i) Chalk ratio 10:3:12. Total 25. Carbon % = 3/25 × 100 = 12%. (ii) Carbon is 3g. 12% of weight = 3g. Weight = 3 × 100/12 = 25g. Question 8: Book CP 275, loss 15%. SP = 85% of 275 = 85/100 × 275 = 233.75 rupees. Question 9: (a) P=1200, R=12%, T=3. I = 1200 × 12 × 3 / 100 = 432. Amount = 1632 rupees. (b) P=7500, R=5%, T=3. I = 7500 × 5 × 3 / 100 = 1125. Amount = 8625 rupees. Question 10: Rate gives 280 interest on 56000 in 2 years. 280 = 56000 × R × 2 / 100. 280 = 1120R. R = 0.25%. Question 11: Interest 45 for one year at 9%. 45 = P × 9 × 1 / 100. P = 45 × 100/9 = 500 rupees.
Finally, let us review What Have We Discussed. First, a way of comparing quantities is percentage. Percentages are numerators of fractions with denominator 100. Percent means per hundred. For example, 82% marks means 82 marks out of hundred. Second, fractions can be converted to percentages and vice versa. For example, 1/4 = 1/4 × 100%, whereas 75% = 75/100 = 3/4. Third, decimals too can be converted to percentages and vice versa. For example, 0.25 = 0.25 × 100% = 25%. Fourth, percentages are widely used in daily life. We have learnt to find exact number when a certain percent of total quantity is given. When parts are given as ratios, we convert to percentages. Increase or decrease in quantity is expressed as percentage. Profit or loss is expressed in percentages. Fifth, while computing interest, rate is given in per cents. For example, 800 rupees borrowed for 3 years at 12% per annum.
Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]